Steady State Diffusion

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In steady state, there is no change in concentration so ∂c/∂t = 0 and Fick's second law says 0 = ∇⋅J. For an inert flat layer, thickness L, this means a linear profile and a flux directly proportional to the concentration difference Δc across the layer:

J = D Δc
L
which is used in experiments to measure diffusion parameters.

In biology, there can be chemical conversion, and steady state implies that this process has to be matched by supply. For tissue with an oxygen (O2) consumption M, the changes due to consumption M and diffusion – Fick's second law – must cancel out:

0 = − M − ∇⋅J
and combination with Fick's first law leads to:
0 = − M + 2P
where now the form with partial pressure P is used because it is the gas oxygen that has to supply the tissue. Again, for a flat layer:
J = ℘ ΔP + ML(½L − x)
L
But more interesting is the famous Krogh-Erlang equation for O2 diffusing out of a central capillary into a surrounding circular (2-dimensional) region:
P = Pc  + M { r2 − rc2 − R2 ln( r2 )} 
4 rc2
where Pc is capillary rim O2 pressure, r distance from the center, where the capillary is, with radius rc, and R radius of the circular region. For more about that choose the ‘oxygen transport in muscle tissue’ item in the “Career” page.

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